prove that A,Q, B and T lie on a circle

Question

two circles intersect in points P and Q. a secant passing through P intersects the circles in A and B respectively. tangents to the circles at A and B intersect at T

Answer 1

Find the perpendicular bisectors of segments AQ, QB, BT and TA.  If a point lies on the perpendicular bisector of a segment, all points on that line are equidistant from the endpoints.  Therefore, the point at which these lines meet is the center of a circle.

 

That is a nice, interesting problem.